Download - xOne Port Networks
-
8/2/2019 xOne Port Networks
1/53
i-v Characteristics of
One-Port Networks
Topic 2
-
8/2/2019 xOne Port Networks
2/53
The concept of a v-i characteristic need not be confined to anindividual element.
A group of elements connected together can be described by asingle equation (characteristic curve) that relates the voltage and
current variables at any one port.
Such a port can be considered as a single element with a specificv-i characteristic.
The composite characteristics of series- or parallel-connected
elements can be obtained by the method of graphical addition,except that we need to take into account the reference directionsof unilateral elements when drawing their characteristics.
Composite v-i characteristics
-
8/2/2019 xOne Port Networks
3/53
Consider the circuit shown in Figure 1 where two two-terminal elementsare connected in series at node B. Nodes A and C are connected to themeasuring circuit comprising a variable voltage source, an ammeter, anda voltmeter.
Finding the Composite v-i Characteristics of Series-Connected Elements
A
C
v
i
Variable
voltage
source
Circuit
element 1
Circuit
element 2
B
v1
v2
#2
#1
Figure 1
-
8/2/2019 xOne Port Networks
4/53
The one-port, consisting of elements 1 and 2, whose terminals arenodes A and C, is called the series connection of elements 1 and 2.
The two elements #1 and #2 are specified by their characteristics,which are assumed known.
A
C
Circuit
element 1
Circuit
element 2
B
v1
v2 #2
#1
1-port netwok formed by
series connection of
elements #1 and #2.
Figure 2
-
8/2/2019 xOne Port Networks
5/53
To find the composite characteristic of the series connected elements,we note that the same current i flows through both elements. Theterminal voltage v is given by the sum of voltage drops v1 and v2. Thatis,
i = i1 = i2
andv = v1 + v2
A
C
v
Circuit
element 1
Circuit
element 2
B
v1
v2
Figure 3
-
8/2/2019 xOne Port Networks
6/53
Thus, the composite characteristic can be obtained as follows:
Step 1. Select suitable values of current on the the i-v characteristiccurve of each element and draw horizontal lines passingthrough these selected current values to intersect the i-v
curve.
Step 2. Add the two curves on a point-by-point basis at eachselected values of current. The result is a combined curvefor the two elements.
-
8/2/2019 xOne Port Networks
7/53
ExampleObtain the composite characteristics of the one-port network shown inFigure 4, which consist of a linear resistor connected in series with anideal diode (a unilateral circuit element).
1
Ideal
diode
v
i
Figure 4
-
8/2/2019 xOne Port Networks
8/53
Step 1. Define the reference directions of the terminal voltage andterminal current of each circuit element.
For series-connected elements,
i. use the reference direction of the input current to define
the reference direction of the terminal current of eachelement.
ii. place the positive and negative signs of the terminalvoltage of each circuit element such that they are on thesame side as that of the input voltage.
Solution
i
1 v1
i
Ideal
diodev2
v
v1
i
Ideal
diodev2
1
Figure 5
-
8/2/2019 xOne Port Networks
9/53
Step 2. Draw on the same graph the i-v characteristic curve of eachcircuit element based on their reference directions defined inStep 1.
Solution (continued)
v
i
1 A
1 V
0
1- resistor
Ideal diode
Figure 6
-
8/2/2019 xOne Port Networks
10/53
Step 3. Add, at suitably selected values of terminal current, thecorresponding terminal voltage of each circuit element.
Step 4. Draw on a graph of i versus v, the composite i-v characteristiccurve of the one-port using the pairs of values of terminalcurrent and terminal voltage obtained in Step 3.
Figure 7 shows a plot of the composite characteristic curve forthe one-port circuit based on the above steps.
Solution (continued)
v
i
1 A
1 V
0
Composite
characteristic
curve
Figure 7
-
8/2/2019 xOne Port Networks
11/53
ExampleObtain the composite characteristics of the one-port network shown inFigure 8, which consist of an resistor (a linear resistor) connected inseries with an ideal diode (a unilateral circuit element).
1
Ideal
diode
v
i
Figure 8
-
8/2/2019 xOne Port Networks
12/53
Step 1. Define the reference directions for the terminal voltage andterminal current of each circuit element.
For series-connected elements,
i. use the reference direction of the input current to define
the reference direction of the terminal current of eachelement.
ii. place the positive and negative signs of the terminalvoltage of each circuit element such that they are on thesame side as that of the input voltage.
Solution
i
1 v1
i
Ideal
diodev2
v
v1
i
Ideal
diodev2
1
Figure 9
-
8/2/2019 xOne Port Networks
13/53
Step 2. Draw on the same graph the i-v characteristic curve of eachcircuit element based on their reference directions defined inStep 1.
Solution (continued)
v
i
1 A
1 V
0
1- resistor
Ideal diode
Figure 10
-
8/2/2019 xOne Port Networks
14/53
Step 3. Add, at suitably selected values of terminal current, thecorresponding terminal voltage of each circuit element.
Step 4. Draw on a graph of i versus v, the composite i-v characteristiccurve of the one-port using the pairs of values of terminalcurrent and terminal voltage obtained in Step 3.
Figure x shows a plot of the composite characteristic curve forthe one-port circuit based on the above steps.
Solution (continued)
v
i
1 A
1 V
0
Composite
characteristic
curve
-1 V
- 1 A
Figure 11
-
8/2/2019 xOne Port Networks
15/53
Example 3Obtain the composite characteristics of the one-port network shown inFigure 12.
1
Ideal
diodev
i
2 V
Figure 12
-
8/2/2019 xOne Port Networks
16/53
Step 1. Define the reference directions for the terminal voltage andterminal current of each circuit element.
For series-connected elements,
i. use the reference direction of the input current to define
the reference direction of the terminal current of eachelement.
ii. place the positive and negative signs of the terminalvoltage of each circuit element such that they are on thesame side as that of the input voltage.
Solution
-
8/2/2019 xOne Port Networks
17/53
i
v1
i
Ideal
diodev2
1 i
1
v
v3 1 V
v1
v2
v3 1 V
i
Figure 13
-
8/2/2019 xOne Port Networks
18/53
Step 2. Draw on the same graph the i-v characteristic curve of eachcircuit element based on their reference directions defined inStep 1.
Solution (continued)
v
i
1 A
1 V battery
0
1- resistor
Ideal diode
Figure 14
-
8/2/2019 xOne Port Networks
19/53
Step 3. Add, at suitably selected values of terminal current, thecorresponding terminal voltage of each circuit element.
Step 4. Draw on a graph of i versus v, the composite i-v characteristiccurve of the one-port using the pairs of values of terminalcurrent and terminal voltage obtained in Step 3.
Figure 15 shows a plot of the composite characteristic curve forthe one-port circuit based on the above steps.
Solution (continued)
v
i
1 A
1 V
0
Composite
characteristiccurve
- 1 A
2 VFigure 15
-
8/2/2019 xOne Port Networks
20/53
Example 4Obtain the composite characteristics of the one-port network shown inFigure 16.
Ideal zener diode
VZK = 5.6 V
v
i
ZD1
ZD2 Ideal zener diode
VZK = 9.4 V
Figure 16
-
8/2/2019 xOne Port Networks
21/53
Step 1. Define the reference directions of the terminal voltage andterminal current of each circuit element.
For series-connected elements,
i. use the reference direction of the input current to define
the reference direction of the terminal current of eachelement.
ii. place the positive and negative signs of the terminalvoltage of each circuit element such that they are on thesame side as that of the input voltage.
Solution
-
8/2/2019 xOne Port Networks
22/53
v
VZK = 9.4 V
VZK = 5.6 V
i
ZD1v1
ZD2v2
VZK = 5.6 V
i
ZD1v1
VZK = 9.4 V
i
ZD2v2
Figure 17
-
8/2/2019 xOne Port Networks
23/53
Step 2. Draw on the same graph the i-v characteristic curve of eachcircuit element based on their reference directions defined inStep 1.
Solution (continued)
v
i
0
ZD1
-5.6 V
Figure 18
-
8/2/2019 xOne Port Networks
24/53
Step 2. Draw on the same graph the i-v characteristic curve of eachcircuit element based on their reference directions defined inStep 1.
Solution (continued)
v
i
9.4 V
0
ZD2
Figure 18
-
8/2/2019 xOne Port Networks
25/53
Step 3. Add, at suitably selected values of terminal current, thecorresponding terminal voltages of each circuit element.
Step 4. Draw on a graph of i versus v, the composite i-v characteristiccurve of the one-port using the pairs of values of terminalcurrent and terminal voltage obtained in Step 3.
Figure x shows a plot of the composite characteristic curve forthe one-port circuit based on the above steps.
Solution (continued)
v
i
-5.6 V
0
Compositecharacteristic
curve
9.4 V
Figure 19
-
8/2/2019 xOne Port Networks
26/53
Finding the Composite characteristic curve of parallel-connected elements
Any two elements are called in parallel if and only if:
i. One terminal of each element is connected to a common node
ii. The other terminal of each element is connected to another commonnode.
When any two elements are connected in parallel, it follows that the voltageacross one is identical to the voltage across the other. Also, the totalcurrent of the combination is equal to the sum of the two individualelements.
i = i1 + i2
v = v1 = v2#1 #2
i
i1 i2
v
A
B
v1 v2
Figure 20
-
8/2/2019 xOne Port Networks
27/53
Worked ExampleDraw the composite characteristics for the parallel-connected solarcells shown in Figure 21a. The solar cells are identical and their i-vcharacteristic is shown in Figure 21b. Determine its maximum outputvoltage and maximum output current.
i
v
(a)
v
i
1 V
-10 mA
0
i
v
(b)Figure 21
-
8/2/2019 xOne Port Networks
28/53
For the parallel connected solar cells, the terminal current is the sumof the currents contributed by each cell; that is
i = i1 + i2 + i3 + i4
At v = 0 V,
i1(max) = i2(max) = i3(max) = i4(max) = 10 mA
Hence, at v = 0 V
i(max)
= i1(max)
+ i2(max)
+ i3(max)
+ i4(max)
= 10 mA + 10 mA + 10 mA + 10 mATherefore,
i(max) = 40 mA
Solution
-
8/2/2019 xOne Port Networks
29/53
v
i
1 V
40 mA
0
i
v
Solution (continued)
The composite i-v characteristic is as shown in Figure 22b.
Figure 22
(a)
(b)
-
8/2/2019 xOne Port Networks
30/53
Drill ExerciseDraw the composite characteristics for the circuit shown in Figure 23.
3 2 v
iA
B
Figure 23
-
8/2/2019 xOne Port Networks
31/53
Composite characteristic of series-parallel elements
#1
A
B
#2
#3
The composite characteristic of a 1-port comprising of series- and parallel-connected elements can be found in a few steps by replacing eachparallel and series-connected elements with their circuit equivalents.
Figure 24
C
-
8/2/2019 xOne Port Networks
32/53
#1
i
i1
A
v1 #2
i2
v2va
C
A
C
#A
-
8/2/2019 xOne Port Networks
33/53
A
B
v
Circuit
element A
Circuit
element 3
C
va
v3
#1
A
B
#2
#3
C
-
8/2/2019 xOne Port Networks
34/53
#1
i
i1
v
A
B
v1 #2
i2
v2
#3
i3
v3
-
8/2/2019 xOne Port Networks
35/53
R1= 1
v
i
L1 L2
10
v (V)
i (A)
00
10 20
i
v 5
Example
Using the circuit of Figure xa and the incandescent lamp characteristicshown in Figure xb, obtain by graphical method the driving pointcharacteristic of the circuit. Assume the two incandescent lamps haveidentical i-v chacteristics.
-
8/2/2019 xOne Port Networks
36/53
Solution
The incandescent lamps are bilateral devices and their voltage andcurrent reference directions are immaterial in this problem. Thus, wecan simply proceed to do graphical addition in a straight-forwardmanner.
i = i1 + i2
v = v1 = v2#1 #2
i
i1 i2
v
A
B
v1 v2
Step 1. Use graphical addition of parallel-connected elements tofind the equivalent i-v characteristic of the parallel-connectedlamps.
-
8/2/2019 xOne Port Networks
37/53
v
VZK = 9.4 V
VZK = 5.6 V
i
ZD1v1
ZD2v2
VZK = 5.6 V
i
ZD1v1
VZK = 9.4 V
i
ZD2v2
Step 3. Use graphical addition of series-connected elements
to obtain the composite characteristic.
-
8/2/2019 xOne Port Networks
38/53
1 k
RL= 1 kv
i
v
i
i
v- 5 V
ExampleConsider the one-port network shown in Figure x. The zener diode ismodeled by a three-segment piecewise-linear characteristic shownin Figure xb. Using the graphic method only, find the driving pointcharacteristic of the one-port network.
-
8/2/2019 xOne Port Networks
39/53
Drill ExerciseAssuming an ideal zener diode characteristic with VZ = 5 V, find by using thegraphical method the driving point characteristic of the one-port N shown inFigure x.
1 k
E = 10 V
i
v
-
8/2/2019 xOne Port Networks
40/53
Composite characteristic of linear One-Port networksFor linear one port networks, it is possible to obtain the i-v characteristicentirely by analytical method. The following examples illustrates the idea.
3
3 volts
v
iA
B
Worked Example
Figure x show a one-port network consisting of two linear elementsconnected in series. Draw the driving point characteristics for thecircuit shown in Figure x.
-
8/2/2019 xOne Port Networks
41/53
ExampleFigure x show a one-port network consisting of two linear elementsconnected in parallel. Derive the equation describing the following one-port in terms of the port voltage and port current and plot it graphicallyin the i-v plane.
1 A2 v
iA
B
-
8/2/2019 xOne Port Networks
42/53
2
4
6
8
10
i (ma)
v (V)0 1 2 3 4 5 6 7 8
6.67 V
i-v characteristic
of the 1-port
network
-
8/2/2019 xOne Port Networks
43/53
2
4
6
8
10
i (ma)
v (V)0 1 2 3 4 5 6 7 8
6.67 V
i-v characteristic
of the 1-port
network
-
8/2/2019 xOne Port Networks
44/53
For the circuits shown in Figure S2,i. derive the equation describing the one-port in terms of the port
voltage and the port current.ii. Plot the i-v characteristic in the i-v plane.
Example
10 V
A
1 k
2 k v
i
B
-
8/2/2019 xOne Port Networks
45/53
10 V
A
1 k
2 k
i2
v
i
B
i1
Step 2. Derive the circuits current equation by applying KCL at node A.
21 iii
Solution
Step 1. Label all branch currents.
Step 3. Apply Ohms law to the 2 k and 1 k resistors.
1000
10
1
1
v
R
vEi
20002
2
v
R
vi
-
8/2/2019 xOne Port Networks
46/53
Step 4. Substitute Eqs.() and () into Eq.().
20001000
1021
vviii
2000
20
2000
3
vi
Step 5. Simplify expression.
Upon simplifying Eq.(), we obtain the required i-v characteristicequation of the one-port as follows:
-
8/2/2019 xOne Port Networks
47/53
Step 6. Plot the i-v characteristic equation in the i-v plane.
We note that since Eq.() represents that of a straight line, we caneasily plot the characteristic equation by simply finding its x- and y-intercepts and then connecting the two points on the graph with a
straight line.
Coordinates of the x-intercept is obtained by setting v = 0 in Eq.().Thus,
mA102000
20
0
vi
Similarly, coordinates of the y-intercept is obtained by setting i = 0in Eq.(). Thus,
V67.63
20
0
iv
-
8/2/2019 xOne Port Networks
48/53
2
4
6
8
10
i (ma)
v (V)
0 1 2 3 4 5 6 7 8
6.67 V
i-v characteristic
of the 1-portnetwork
A plot of the i-v characteristic of the one-port is shown in Figure x.
-
8/2/2019 xOne Port Networks
49/53
The one-port network shown in Figure x contains both ac and dcvoltage sources.
i. Derive the i-v characteristic of the network for the given voltageand current reference directions.
ii. Plot the i-v characteristic in the i-v plane.
Example
1 k
vs(t) = 10sin 10t
i
v
-
8/2/2019 xOne Port Networks
50/53
Solution
Step 1. Apply KVL to the circuit to obtain an equation that relates
the terminal voltage and the terminal current to the circuit elements.
0510sin10 itv
or
tvi 10sin105
1
Step 2. Plot the i-v characteristic in the i-v plane.
We note that Eq() is that of a straight line with slope -5 and ay-intercept that is a function of time.
-
8/2/2019 xOne Port Networks
51/53
2
4
6
8
10
i (A)
v (V)0 20 40 60
i-v characteristic of the 1-port
network at t = 3/20 s.
-10
- 8
- 6
- 4
- 2
-20-40
vi5
1(i) At t = 0,
-
8/2/2019 xOne Port Networks
52/53
2
4
6
8
10
i (A)
v (V)0 20 40 60
i-v characteristic of the 1-portnetwork at t = /20 s.
-10
- 8
- 6
- 4
- 2
-20-40
105
1 vi(ii) At t = /20,
-
8/2/2019 xOne Port Networks
53/53
2
4
6
8
10
i (A)
v (V)0 20 40 60
i-v characteristic of the 1-port
network at t = 3/20 s.
- 8
- 6
- 4
- 2
-20-40
105
1 vi(iii) At t = 3/20,